A n Introductio n to Measur e an d Integratio n

http://dx.doi.org/10.1090/gsm/045

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A n Introductio n to Measur e an d Integratio n SECOND EDITION

Inder K. Rana

Graduate Studies

in Mathematics

Volume 45

;S!HSSJ»0

American Mathematical Society Providence, Rhode Island

Editorial Board

Steven G. Krantz David Saltman (Chair)

David Sattinger Ronald Stern

2000 Mathematics Subject Classification. P r imary 28-01; Secondary 28A05, 28A10, 28A12, 28A15, 28A20, 28A25,

28A33, 28A35, 26A30, 26A42, 26A45, 26A46.

ABSTRACT. This text presents a motivated introduction to the theory of measure and integration. Starting with an historical introduction to the notion of integral and a preview of the Riemann integral, the reader is motivated for the need to study the Lebesgue measure and Lebesgue integral. The abstract integration theory is developed via measure. Other basic topics discussed in the text are Pubini's Theorem, Lp-spaces, Radon-Nikodym Theorem, change of variables formulas, signed and complex measures.

Library of Congress Cataloging-in-Publication D a t a

Rana, Inder K. An introduction to measure and integration / Inder K. Rana.—2nd ed.

p. cm. — (Graduate texts in mathematics, ISSN 1065-7339 ; v. 45) Includes bibliographical references and index. ISBN 0-8218-2974-2 (alk. paper) 1. Lebesgue integral. 2. Measure theory. I. Title. II. Graduate texts in mathematics ; 45.

QA312 .R28 2002 515/.42—dc21 2002018244

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publica­tion is permitted only under license from Narosa Publishing House. Requests for such permis­sion should be addressed to Narosa Publishing House, 6 Community Centre, Panchscheel Park, New Delhi 110 017, India.

First Edition © 1997 by Narosa Publishing House. Second Edition © 2002 by Narosa Publishing House. All rights reserved.

Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02

In memory of my father

Shri Omparkash Rana

(24th April, 1924-26th May, 2002)

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Contents

Preface xi

Preface to the Second Edition xvii

Recipe for a one semester course and interdependence of the chapters xix

Notations used in the text xxi

Prologue. The length function 1

Chapter 1. Riemann integration 5

§1.1. The Riemann integral: A review 5

§1.2, Characterization of Riemann integrable functions 18

§1.3. Historical notes: The integral from antiquity to Riemann 30

§1.4. Drawbacks of the Riemann integral 36

Chapter 2. Recipes for extending the Riemann integral 45

§2.1. A function theoretic view of the Riemann integral 45

§2.2. Lebesgue's recipe 47

§2.3. Riesz-Daniel recipe 49

Chapter 3. General extension theory 51

§3.1. First extension 51

§3.2. Semi-algebra and algebra of sets 54

vm Contents

§3.3. Extension from semi-algebra to the generated algebra 58

§3.4. Impossibility of extending the length function to all subsets of

the real line 61

§3.5. Countably additive set functions on intervals 62

§3.6. Countably additive set functions on algebras 64

§3.7. The induced outer measure 70

§3.8. Choosing nice sets: Measurable sets 74

§3.9. The cr-algebras and extension from the algebra to the generated

cr-algebra 80

§3.10. Uniqueness of the extension 84

§3.11. Completion of a measure space 89

Chapter 4. The Lebesgue measure on R and its properties 95

§4.1. The Lebesgue measure 95

§4.2. Relation of Lebesgue measurable sets with topologically nice subsets of R 99

§4.3. Properties of the Lebesgue measure with respect to the group

structure on R 103

§4.4. Uniqueness of the Lebesgue measure 106

§4.5. * Cardinalities of the a-algebras C and B^ HO

§4.6. Nonmeasurable subsets of R 113

§4.7. The Lebesgue-Stieltjes measure 114

Chapter 5. Integration 117

§5.1. Integral of nonnegative simple measurable functions 118

§5.2. Integral of nonnegative measurable functions 122

§5.3. Intrinsic characterization of nonnegative measurable functions 130

§5.4. Integrable functions 143

§5.5. The Lebesgue integral and its relation with the Riemann

integral 153

§5.6. I/i[a, b] as completion of 1Z[a, b] 158

§5.7. Another dense subspace of L\[a, b] 163

§5.8. Improper Riemann integral and its relation with the Lebesgue integral 168

Contents IX

§5.9. Calculation of some improper Riemann integrals 172

Chapter 6. Fundamental theorem of calculus for the Lebesgue integral 175

§6.1. Absolutely continuous functions 175

§6.2. Differentiability of monotone functions 179

§6.3. Fundamental theorem of calculus and its applications 191

Chapter 7. Measure and integration on product spaces 209

§7.1. Introduction 209

§7.2. Product of measure spaces 212

§7.3. Integration on product spaces: Fubini's theorems 221

§7.4. Lebesgue measure on M2 and its properties 229

§7.5. Product of finitely many measure spaces 237

Chapter 8. Modes of convergence and Lp-spaces 243

§8.1. Integration of complex-valued functions 243

§8.2. Convergence: Point wise, almost everywhere, uniform and

almost uniform 248

§8.3. Convergence in measure 255

§8.4. Lp-spaces 261

§8.5. *Necessary and sufficient conditions for convergence in Lp 270

§8.6. Dense subspaces of Lp 279

§8.7. Convolution and regularization of functions 281

§8.8. LOO(X,S,IJ,): The space of essentially bounded functions 291

§8.9. 1/2(X, 5,/i): The space of square integrable functions 296

§8.10. Z/2-convergence of Fourier series 306

Chapter 9. The Radon-Nikodym theorem and its applications 311

§9.1. Absolutely continuous measures and the Radon-Nikodym

theorem 311

§9.2. Computation of the Radon-Nikodym derivative 322

§9.3. Change of variable formulas 331

Chapter 10. Signed measures and complex measures 345

X Contents

§10.1. Signed measures 345

§10.2. Radon-Nikodym theorem for signed measures 353

§10.3. Complex measures 365

§10.4. Bounded linear functionals on Lrp(X, 5 , /i) 373

Appendix A. Extended real numbers 385

Appendix B. Axiom of choice 389

Appendix C. Continuum hypothesis 391

Appendix D. Urysohn's lemma 393

Appendix E. Singular value decomposition of a matrix 395

Appendix F. Functions of bounded variation 397

Appendix G. Differentiable transformations 401

References 409

Index 413

Index of notations 419

Preface

"Mathematics presented as a closed, linearly ordered, system of truths without reference to origin and purpose has its charm and satisfies a philosophical need. But the attitude of introverted science is un­suitable for students who seek intellectual independence rather than indoctrination; disregard for applications and intuition leads to isola­tion and atrophy of mathematics. It seems extremely important that students and instructors should be protected from smug purism."

Richard Courant and Fritz John (Introduction to Calculus and Analysis)

This text presents a motivated introduction to the subject which goes under various headings such as Real Analysis, Lebesgue Measure and Integration, Measure Theory, Modern Analysis, Advanced Analysis, and so on.

The subject originated with the doctoral dissertation of the French mathematician Henri Lebesgue and was published in 1902 under the ti­tle Integrable, Longueur, Aire. The books of C. Caratheodory [8] and [9], S. Saks [35], LP. Natanson [27] and P.R. Halmos [14] presented these ideas in a unified way to make them accessible to mathematicians. Because of its fundamental importance and its applications in diverse branches of mathe­matics, the subject has become a part of the graduate level curriculum.

Historically, the theory of Lebesgue integration evolved in an effort to remove some of the drawbacks of the Riemann integral (see Chapter 1). However, most of the time in a course on Lebesgue measure and integra­tion, the connection between the two notions of integrals comes up only

XI

Xl l Preface

after about half the course is over (assuming that the course is of one se­mester). In this text, after a review of the Riemann integral, the reader is acquainted with the need to extend it. Possible methods to carry out this extension are sketched before the actual theory is presented. This approach has given satisfying results to the author in teaching this subject over the years and hence the urge to write this text. The nucleus for the text was provided by the lecture notes of the courses I taught at Kurukshetra Uni­versity (India), University of Khartoum (Sudan), South Gujarat University (India) and the Indian Institute of Technology Bombay (India). These notes were slowly augmented with additional material so as to cover topics which have applications in other branches of mathematics. The end product is a text which includes many informal comments and is written in a lecture-note style. Any new concept is introduced only when it is needed in the logical development of the subject and it is discussed informally before the exact definition appears. The subject matter is developed by motivating examples and probing questions, as is normally done while teaching. I have tried to avoid slick proofs. Often a proof is either divided into steps or is presented in such a way that the main ideas of the proof emerge before the details follow.

Summary of the text

The text opens with a Prologue on the length function and its properties which are basic for the development of the subject.

Chapter 1 begins with a detailed review of the Riemann integral and its properties. This includes Lebesgue's characterization of Riemann integrable functions. It is followed by a brief discussion on the historical development of the integral from antiquity (around 300 B.C.) to the times of Riemann (1850 A.D.). For a detailed account the reader may refer to Bourbaki [6], Hawkins [16] and Kline [20]. The main aim of the historical notes is to make young readers aware of the fact that mathematical concepts arise out of physical problems and that it can take centuries for a concept to evolve. This section also includes Riemann's example of an integrable function having an infinite number of discontinuities and a proof of the fundamental theorem of calculus due to G. Darboux. The next section of Chapter 1 has a discussion about the drawbacks of the Riemann integral, including the example due to Vito Volterra (1881) of a differentiable function / : [0,1] —> R whose derivative function is bounded but is not Riemann integrable. These considerations made mathematicians look for an extension of the Riemann integral and eventually led to the construction of the Lebesgue integral.

Chapter 2 discusses two significantly different approaches for extending the notion of the Riemann integral. The one due to H. Lebesgue is sketched

Preface xin

in this chapter and is discussed in detail in the rest of the book. The second is due to P.J. Daniel [10] and F. Riesz [32], an outline of which is given. These discussions motivate the reader to consider an extension of the length function from the class of intervals to a larger class of subsets of EL

Chapters 3, 4 and 5 form the core of the subject: extension of measures and the construction of the integral in the general setting with the Lebesgue measure and the Lebesgue integral being the motivating example. The pro­cess of extension of additive set functions (known as the Caratheodory ex­tension theory) is discussed in the abstract setting in Chapter 3. This chapter also includes a result due to S.M. Ulam [41] which rules out the possibility of extending (in a meaningful way) the length function to all subsets of R, under the assumption of the Continuum Hypothesis.

The outcomes of the general extension theory, as developed in Chapter 3, are harvested for the particular case of the real line and the length function in Chapter 4. This gives the required extension of the length function, namely, the Lebesgue measure. Special properties of the Lebesgue measure and Lebesgue measurable sets (the collection of sets on which the Lebesgue measure is defined by the extension theory) with respect to the topology and the group structure on the real line are discussed in detail. The final section of the chapter includes a discussion about the impossibility of extending the length function to all subsets of E under the assumption of the Axiom of Choice.

In Chapter 5, the construction of the extended notion of integral is discussed. Once again, the motivation comes from the particular case of functions on the real line. Lebesgue's recipe, as outlined in Chapter 2, is carried out for the abstract setting. The particular case gives the required integral, namely, the Lebesgue integral. The space Li[a,fe] of Lebesgue in-tegrable function on an interval [a, b] is shown to include TZ[a, 6], the space of Riemann integrable functions, the Lebesgue integral agreeing with the Riemann integral on lZ[a. b]. Also, it is shown that L\[a.7 b] is the completion of 1Z[a, b] under the Li-metric. The final section of the chapter discusses the relation between the Lebesgue integral and the improper Riemann integral.

Chapter 6 gives a complete proof of the fundamental theorem of cal­culus for the Lebesgue integral. (This theorem characterizes the pair of functions F, / such that F is the indefinite integral of / . This removes one of the main drawbacks of Riemann integration.) As applications of the fun­damental theorem of calculus, the chain-rule and integration by substitution for the Lebesgue integral are discussed.

The remaining chapters of the book include special topics. Chapter 7 deals with the topic of measure and integration on product spaces, with

XIV Preface

Fubini's theorem occupying the central position. The particular case of Lebesgue measure on R2 and its properties are discussed in detail.

Chapter 8 starts with extending the concept of integral to complex valued functions. The remaining sections discuss various methods of ana­lyzing the convergence of sequences of measurable functions. The Lp-spaces and discussion of some of their dense subspaces in the special case of the Lebesgue measure space are also included in this chapter. The last section of the chapter includes a brief discussion on the application of Lebesgue integration to Fourier series.

Chapter 9 includes a discussion of the Radon-Nikodym theorem. As an application, the change of variable formulas for Lebesgue integration on Rn are derived.

In Chapter 10, the additive set functions, which are not necessarily nonnegative or even real-valued, are discussed. The main aim is to prove the Hahn decomposition theorem and the Lebesgue decomposition theorem. As a consequence, an alternative proof of the Radon-Nikodym theorem is given. This chapter also includes a discussion of complex measures.

The text has three appendices. Appendix E gives a proof of the sin­gular value decomposition of matrices, needed in sections 7.4 and 9.3. In Appendix F, functions of bounded variation (needed in section 6.1) are discussed. Appendix G includes a discussion of differentiate transforma­tion and a proof of the inverse function theorem, needed in section 9.3. (In the present edition four more appendices, A,B,C and D, have been added.)

The text is sprinkled with 200 exercises, most of which either include a hint or are broken into doable steps. Exercises marked with • are needed in later discussions. The sections and exercises marked with * can be omitted on first reading. Some of the results in the text are credited to the discoverer, but no effort is made to trace the origin of each result. In any case, no originality is claimed.

Prerequisites and course plans

The text assumes that the reader has undergone a first course in mathe­matical analysis (roughly equivalent to that of first five chapters of Apostol [2]). The text as such can be used for a one-year course. A recipe for a one-semester course (approximately 40 lecture hours and 10 problem discus­sion hours) on Lebesgue measure and integration is given after the preface. Since the text is in a lecture-note style, it is also suitable for an individ­ual self-study program. For such readers, the chart depicting the logical interdependence of the chapters will be useful.

Preface xv

Acknowledgments

It is difficult to list all the individuals and authors who have influenced and helped me in preparing this text, directly or indirectly. First of all I would like to thank my teacher and doctoral thesis advisor, Prof. K.R. Parthasarathy (Indian Statistical Institute, Delhi), whose lectures at the University of Bombay (Mumbai) and the Indian Statistical Institute (Delhi), clarified many concepts and kindled my interest in the subject. I learned much from his style of teaching and mathematical exposition.

Some of the texts which have influenced me in one form or another are Halmos [14], Royden [34], Hewitt and Stromberg [18], Aliprantis and Burkinshaw [1], Friedman [13] and Parthasarathy [28].

I am indebted to the students to whom I have taught this subject over the years for their reactions, remarks, comments and suggestions which have helped in deciding on the style of presentation of the text.

It is a pleasure to acknowledge the support and encouragement I received from my friend Prof. S. Kumaresan (University of Bombay) at various stages in the preparation of this text. He also went through the text, weeding out misprints and mistakes. I am also thankful to my friend Dr. S. Purkayastha (Indian Institute of Technology Bombay) for going through the typeset man­uscript and suggesting many improvements. For any shortcoming still left in the text, the author is solely responsible.

I thank C.L. Anthony for processing the entire manuscript in I TfrjX. The hard job of preparing the figures was done by P. Devaraj, I am thankful for his help. Thanks are also due to the Department of Mathematics, IIT Bombay for the use of Computer Lab and photocopying facilities. I would like to thank the Curriculum Development Program of the Indian Institute of Technology Bombay for the financial support to prepare the first version of the manuscript. The technical advice received from the production de­partment of Narosa Publishers in preparing the camera ready copy of the manuscript is acknowledged with thanks.

Special thanks are due to my family: my wife Lalita for her help in more ways than one; and my parents for allowing me to choose my career and for their love and encouragement in pursuing the same. It is to them that this book is dedicated.

Finally, I would be grateful for critical comments and suggestions for later improvements.

Mumbai , 1997

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Preface to the Second Edition

In revising the first edition, I have resisted the temptation of adding more topics to the text. The main aim has been to rectify the defects of the first edition:

• Efforts have been made to remove the typos and correct the mis­matched cross references. I hope there are none now.

• In view of the feedback received from students, at many places phrases like 'trivial to verify', 'easy to see', etc have been expanded with explanations.

• Sequencing of topics in some of the chapters has been altered to make the development of the subject matter more consistent.

• Short notes have been added to give a glimpse of the link between measure theory and probability theory.

• More exercises have been added.

• Four new appendices have been added.

While preparing the first edition of this book, I was often questioned about the 'utility' of spending my valuable 'research time' on writing a book. The response of the students to the first edition and the reviewers' comments have confirmed my confidence that writing a book is as valuable as doing research. I thank all the reviewers of the first edition for their encouraging remarks. Their constructive criticism has helped me a lot in preparing this edition.

xvn

XV111 Preface to the Second Edition

I would like to thank Mr. N. K. Mehra, Narosa Publishers, for agreeing to copublish this edition with the AMS.

I take pleasure in offering thanks to Edward G. Dunne, Acquisitions Editor, Book Program AMS, for the help and encouragement received from him.

I thank the editorial and the technical support staff of the AMS for their help and cooperation in preparing this edition.

Once again, the help received from P. Devraj in revising the figures is greatly appreciated. Thanks are also due to Mr. C.L. Anthony and Clarity Reprographers & Traders for typesetting the manuscript in I^TEK.

I would greatly appreciate comments/suggestions from students and teachers about the present edition. I intend to post comments/corrections on the present edition on my homepage at

www.math.iitb.ac.in/~ikr/books.html

Mumbai, 2002 Inder K. Rana

Recipe for a one semester course and interdependence of the chapters

*Lebesgue measure and integration (40 lectures and 10 problem/discussion hours)

Prologue: Everything

Chapter 1: Sections 1.1 and 1.2 (depending upon the background of the students), 1.3 and 1.4 can be left for self study.

Chapter 2: Sections 2.1 and 2.2.

Chapter 3: Sections 3.1 to 3.3; 3.5 to 3.9; 3.10 and 3.11 (omitting the proofs).

Chapter 4: Sections 4.1 to 4.3; 4.4 and 4.5 (omitting the proofs); 4.6.

Chapter 5: Sections 5.1 to 5.6; Parts of 5.7 to 5.9 can be included de­pending upon the background of students and the emphasis of the course.

Chapter 6: Sections 6.1; 6.2 (omitting proofs); 6.3 (stating the theorem 6.3.6 and giving applications: 6.3.8, 6.3.10 to 6.3.13, 6.3.16 (omitting proof).

Chapter 7: Sections 7.1 to 7.4.

xix

xx Recipe for a one semester course and interdependence of the chapters

In terdependence of t h e chapters

Chapter 4 The Lebesgue

measure on IR and its properties

Chapter 6 Fundamental theorem of calculus for the Lebesgue integral

Chapter 10 Signed measures and complex measures

Prologue The length function

Chapter 1

Riemann integration

Chapter 2 Recipes for

extending the Riemann integral

Chapter 3 General extension

theory

Chapter 5

Integration

Chapter 7 Measure and Integration

on product spaces

Chapter 3 3.2, 3.6 and 3.11

only

Chapter 5

5.1 to 5.5 only

Chapter 8 Modes of

convergence and Lp-spaces

Chapter 9 Radon-Nikodym theorem and its

applications Chapter 10

10.1 and 10.2 only

Notations used in the text

The three digit system is used to number the definitions, theorems, propo­sitions, lemmas, exercises, notes and remarks. For example, Theorem 3.2.4 is the 4th numbered statement in section 2 of chapter 3.

The symbol • is used to indicate the end of a proof. The symbol A := B or B =: A means that this equality is the definition of A by B. The symbol • before an exercise means that this exercise will be needed in the later discussions. Sections, theorems, propositions, etc., which are marked * can be omitted on first reading.

The phrase "the following are equivalent:" means each of the listed state­ment implies the other. For example in Theorem 1.1.4, it means that each of the statements (i), (ii) and (iii) implies the other.

The notations and symbols used from logic and elementary analysis are as follows:

7 ^

3 V x e A x g A AcB ACB V{X)

implies; gives does not imply implies and is implied by; if and only if there exists for all; for every x belongs to A x does not belong to A A is a proper subset of B A is a subset of B set of all subsets of X

xxi

XXII Notations used in the text

0 A\B AxB

n

n* inf sup

Uu an A Ac

E dE lim sup

lim inf n—>oo

empty set set of elements of A not in B Cartesian product of A and B

Cartesian product of sets X\,... , Xn.

infimum supremum union intersection symmetric difference complement of a set A closure of a set E boundary points of a set E limit superior; upper limit

limit inferior; lower limit

> : / is a function from X into Y and f(x) — y.

the set of natural numbers the set of integers the set of rational numbers the set of real numbers the set of extended real numbers the set of complex numbers n-dimensional Euclidean space absolute value of x

(a, 6), (a, b], [a, b) [a, b], (-00, oc), j ; . ^ ^ ^ R_ (—00, a), (—00, aj, (a, oo), [a, oo) J

{«n}n>i • sequence with n t h term an. (X, d) : a metric space.

For the list of other symbols used in the text, see the symbol index given at the end of the text.

References

[1] Aliprantis, CD. and Burkinshaw, O. Principles of Real Analysis (3rd Edition). Academic Press, Inc. New York, 1998.

[2] Apostol, T.M. Mathematical Analysis. Narosa Publishing House, New Delhi (India), 1995.

[3] Bartle, Robert G. A Modern Theory of Integration, Graduate Studies in Mathemaics, Volume 32, American Mathematicsl Society,Providence, RI, 2001.

[4] Bhatia, Rajendra Fourier Series. Hindustan Book Agency, New Delhi (India), 1993.

[5] Billingsley, Patrick Probability and Measure. 3rd Edition, John Wiley and Sons, New York, 1995.

[6] Bourbaki, N. Integration, Chap. V. Actualites Sci. Indust. 1244. Hermann, Paris, 1956.

[7] Carslaw, H.S. Introduction to the Theory of Fourier's Series and In­tegrals. Dover Publications, New York, 1952.

[8] Caratheodory, C. Vorlesungen iiber Reelle Funktionen. Leipzig, Teub-ner, and Berlin, 1918.

409

410 References

[9] Caratheodory, C. Algebraic Theory of Measure and Integration. Chelse Publishing Company, New York, 1963 (Originally published in 1956).

[10] Daniell, P.J. A general form of integral. Ann. of Math. (2)19 (1919), 279-294.

[11] DePree, Jonn D. and Swartz, Charles W, Introduction to Analysis, John Wiley & Sons Inc., New York, 1988

[12] Fraenkel, A. A.Abstract Set Theory, Fourth Edition, North-Holland, Amsterdam, 1976.

[13] Friedman, A. Foundations of Modern Analysis. Holt, Rinehart and Winston, Inc., New York, 1970.

[14] Halmos, P.R. Measure Theory. Van Nostrand, Princeton, 1950.

[15] Halmos, P.R. Naive Set Theory. Van Nostrand, Princeton, 1960.

[16] Hawkins, T.G. Lebesgue's Theory of Integration: Its Origins and De­velopment. Chalsea, New York, 1979.

[17] Hewitt, E. and Ross, K.A. Abstract Harmonic Analysis, Vol.1. Springer-Verlag, Heidelberg, 1963.

[18] Hewitt, E. and Stromberg, K. Real and Abstract Analysis. Springer-Verlag, Heidelberg, 1969.

[19] Kakutani, S. and Oxtoby, J.C. A non-separable translation invariant extension of the Lebesgue measure space. Ann.of Math. (2) 52 (1950), 580-590.

[20] Kline, M. Mathematical Thoughts from Ancient to Modern Times. Oxford University Press, Oxford, 1972.

[21] Kolmogorov, A.N. Foundations of Probability Theory. Chelsea Pub­lishing Company, New York, 1950.

[22] Korner, T.W. Fourier Analysis. Cambridge University Press, London, 1989.

References 411

[23] Lebesgue, H. Integrale, longueur, aire. Ann. Math. Pura. Appl. (3) 7 (1902), 231-259.

[24] Luxemburg, W.A.J. Arzela's dominated convergence theorem for the Riemann integral Amer. Math. Monthly 78 (1971), 970-979.

[25] McLeod, Robert M., The Generalized Riemann Integral, Carus Mono­graph, No.20, Mathemaical Associaiton of America, Washington, 1980.

[26] Munkres, James E. Topology, 2nd Edition, Prentice Hall, Englewood Cliffs, NJ, 1999.

[27] Natanson, LP. Theory of Functions of a Real Variable. Frederick Un-gar Publishing Co., New York, 1941/1955.

[28] Parthasarathy, K.R. Introduction to Probability and Measure. Macmillan Company of India Ltd., Delhi, 1977.

[29] Parthasarathy, K. R. Probablity Measures on Metric Spaces, Academic Press, New York, 1967.

[30] Rana, Inder K. From Numbers to Analysis, World Scientific Press, Singapore, 1998.

[31] Riesz, F. Sur quelques points de la theorie des fonctions sommables. Comp. Rend. Acad. Sci. Paris 154 (1912), 641-643.

[32] Riesz, F. Sur Vintegrale de Lebesgue. Acta Math. 42 (1920), 191-205.

[33] Riesz, F. and Sz.-Nagy, B. Functional Analysis. Fredrick Ungar Pub­lishing Co., New York, 1955.

[34] Royden, H.L. Real Analysis (3rd Edition). Macmillan, New York, 1963.

[35] Saks, S. Theory of the Integral. Monografje Matematyczne Vol. 7, Warszawa, 1937.

[36] Serrin, J. and Varberg, D.E. A general chain-rule for derivatives and the change of variable formula for the Lebesgue integral. Amer. Math. Monthly 76 (1962), 514-520.

412 References

[37] Solovay, R. A model of set theory in which every set of reals is Lebesgue measurable. Ann. of Math. (2) 92 (1970), 1-56.

[38] Srivastava, S. M. Borel Sets, Springer-Verlag, Heidelberg, 1998.

[39] Stone, M.H. Notes on integration, I-IV. Proc. Natl. Acad. Sci. U.S. 34 (1948), 336-342, 447-455, 483-490; 35 (1949), 50-58.

[40] Titchmarch, E.C. The Theory of Functions. Oxford University Press, Oxford, 1939 (revised 1952).

[41] Ulam, S.M. Zur Masstheorie in der allgemeinen Mengenlehre. Fund. Math. 16 (1930), 141-150.

[42] Zygmund, A. Trigonometric Series, 2 Vols. Cambridge University Press, London, 1959.

Index

Ho, 112

A®B, 210 a.e., 125 absolutely continuous

- complex measure, 366 - function, 176 - measure, 311 - signed measure, 356, 359

aleph-nought, 391 algebra, 55

- generated, 57 almost everywhere, 125 almost uniformly convergent, 250 analytic set, 112 antiderivative, 30 approximate identity, 291 Archemedes, 30 arrangement, 387 Arzela's theorem, 40, 157

BR, 95 3*1,211 £x,8 2 Baire, Rene, 44 Banach

- algebra with identity, 290 - algebra, commutative, 290 - lattice, 365 - spaces, 266

Bernoulli, Daniel, 31 Bessel's inequality, 298, 308 binomial distribution, 69 Borel, Emile, 44 Borel

- measurable function, 227 - subsets, 95, 102

- subse t s of M2, 211 bounded

- convergence theorem, 151 - linear functional, 302, 375 - variation, 397

C, 243 C(R), 162 C[a,b], 41, 160 C°°[a,6], 167 C°°-function, 286 C°°(1R) , 164 C°°(U), 286 C ~ ( t / ) , 286 C 1 -mapping, 403 Co (M"), 295 CC(R), 162 C c (R n ) , 280, 295 C n E, 56 c, 112 XA(x) , 30 Cantor sets, 23, 24 Cantor's ternary set, 25 Carat heodory, Constant in, 44 cardinal number, 391 cardinality

- of a set, 391 - of the continuum, 112, 391

Cartesian product, 389 Cauchy, Augustein-Louis, 31 Cauchy in measure, 259 Cauchy-Schwartz inequality, 264, 296 chain rule, 204 change of variable for Riemann integration,

200, 207 change of variable formula

413

414 Index

- abstract, 332 - linear, 334 - nonlinear, 338

charactristic function, 30 Chebyshev's inequality, 146, 270 choice function, 390 closed subspace, 298 complete measure space, 92 completion of a measure space, 92 complex measure, 365 complex numbers, 243 conditional expectation, 305, 321 conjugate, 374 continuity from above or below, 66 continuum hypothesis, 392 convergence,

- almost everywhere, 248 - almost uniform, 250 - pointwise, 248 - uniform, 248

convergence in - Lp, 267 - measure, 255 - pth mean, 267 - probability, 261

convergent to - +oo, 386 - - o o , 386

convolution, 283 coordinate functions, 401 countable,

- set, 391 - subadditivity, 4

countably - additive, 59 - subadditive, 59

counting measure, 225, 312 critical values, 98 cylinder set, 58 cylindrical coordinates transformation, 344

(Dfi)(x), CD/z)(s), 327 (£>+/)(c), ( D - / ) ( c ) , 180 ( D + / ) ( c ) , (£>-/)(c) , 180

^ W , 320 d[i

Dfj,(x), 324 d'Alembert, Jean, 31 Darboux, Gaston, 34 Darboux's theorem, 18 Denjoy integral, 172 derivative of a measure, 324 Descartes, Rene, 31 det(T), 334 determinant, 334 differentiable, 324, 401 differential, 401

Dini derivatives, 180 Dini, Ulisse, 37 direct substitution, 36 Dirichlet function, 32 Dirichlet, Peter Gustev Lejeune, 32 discrete,

- measure, 68 - probability measure, 68

distribution, - binomial, 68 - discrete probability, 68 - function, 64, 115 - function, probability, 115 - of the measurable function, 142 - Poisson, 68 - uniform, 68

dominated convergence theorem, an exten­sion, 274

Ex, 215 EV, 215 Egoroff's theorem, 249 equicontinuous, 271 equipotent, 391 equivalent, 158 essential supremum, 291 essentially bounded, 291 Euclid, 30 Euler's identity, 310 Euler, Leonhard, 30 example,

- Riemann's, 32 - Vitali's, 113 - Volterra's, 37

extended, - integration by parts, 200 - real numbers, 1, 386

extension, - of a measure, 60, 83, 88 - of the dominated convergence theorem,

274

U*9), 283 </,<?>, 296

f~g, 43 , 158

f-X(C), 57

fn -^ / , 255

fn - ^ / , 248

fn -±> / , 248

fn ^ / , 248

fn ^ / , 250 HC), 57 Fatou's lemma, 140 finite

- additivity property, 1 - set, 391

Index 415

- signed measure, 346 finitely additive, 59 Fourier

- coefficients, 32, 307 - series, 32, 307

Fourier, Joseph, 31 Fubini's theorem, 189, 221, 222 function,

- absolutely continuous, 176 - C ° ° , 286 - characteristic, 30 - choice, 390 - Dirichlet's, 32 - generalized step, 48 - imaginary part of, 244 - indefinite integral of, 175 - indicator, 30 - infinitely differentiable, 163 - integrable, 143, 244, 358 - Lebesgue singular, 181 - length, 1 - Lipschitz, 177 - locally integrable, 286 - measurable, 135 - negative part of , 49 - nonnegative measurable, 118, 123 - nonnegative simple measurable, 118 - of bounded variation, 397 - popcorn, 26 - positive part of, 49, 135 - real part of, 244 - Riemann integrable, 10 - Riemann integral of, 10 - simple, 48 - simple measurable, 135 - step, 47 - support of, 162 - vanishing at infinity, 283 - with compact support, 280

fundamental theorem of calculus, 34, 191, 195, 197

gamma function, 173 gauge integral, 172 generalized Riemann integral, 172 generated,

- algebra, 57 - monotone class, 86 - cr-algebra, 81

graph of the function, 219

Haar measure, 108 Hahn

- decomposition, 350, 373 - theorem, 349

Hankel, Hermann, 36 Heine-Borel theorem, 108

Hilbert space, 297 Holder's inequality, 263

X, 1, 55 J , 55 Jo, 96 Id, 82 I r , 82 I m ( / ) , 243 / fdfj., 123, 244 J sdfi, 119

fZf(x)dx, 10 fz_fdv, 125, 145

fabf(x)dx, 10

Sjf{x)dx , 10

improper Riemann integral, 168, 169 indefinite integral, 175 indicator function, 30 infinitely differentiable functions, 164 inner product, 296 inner regular, 103 integrable, 143, 244, 358 integral, 123, 144, 244

- lower, 10 - of nonnegative simple measurable func­

tion, 119 - over E , 125 - upper, 10

integration - by parts, 36, 199 - by substitution, 206 - of radial functions, 234

intervals, - with dyadic endpoints, 82 - with rational endpoints, 82

inverse - function theorem, 405 - substitution, 36

Jacobian, 405 Jordan, Camille, 44 Jordan decomposition theorem, 350 Jordan's theorem, 398

kernel, 303 Kurzweil-Henstock integral, 172

L, 135 L+, 123 l o , 48 L+ ,118 L J ( X , 5 , M ) , 244

L[oc(Rn) , 286 L i (X ,S , / i ) , 144 Li[a,6], 154 Li-metric, 159

416 Index

Li(E), 154 LJ(X,<S,M+), 358 LJ(X,<S,/i-) , 358 Lp{X,S,fi), 261 Lpifi), 261 CF, 88 £ R , 95 lim inf h j c <£(#), 180 liminf/i | c^(a:), 180 limsup^c<S>(x), 180 l i m s u p ^ <!>(#), 180 Lebesgue

- decomposition theorem, 318 - dominated convergence, 246 - integrable functions, 154 - integral, 154 - measurable sets, 95, 229, 239 - measure, 95, 239 - measure space, 95, 229 - outer measure, 95 - points, 194, 331 - singular function, 181

Lebesgue, Henri, 44 Lebesgue-Stieltjes measure, 115 Lebesgue-Young theorem, 179 left-open, right-closed intervals, 55 Leibniz rule, 36 Leibniz, Gottfried Wilhem, 30 length function, 1

- countable additivity of, 4 - countable subadditivity of, 4 - finite additivity of, 1 - monotonicity property of, 1 - translation invariance of, 4

limit inferior and superior, 386 Lipschitz function, 177 locally integrable, 286 lower,

- derivative, 327 - integral, 10 - left derivative, 180 - left limit, 180 - right derivative, 180 - right limit, 180 - sum, 7 - variation, 351

Luzin theorem, 254

M(X), 260 M(X,S), 371 M(C), 86 Mb(X,S), 362 \x JL v, 319 fi V v, 365 \i A i/, 365 / x T ~ \ 332 /i-null set, 347

/x+, 351 yT, 351 M*,71 MF, 62 mean value property, 156 measurable,

- cover, 90 - function, 49, 135, 260 - kernel, 90 - nonnegative function, 123 - partition, 129, 352 - rectangle, 209, 212 - set, 76 - space, 92 - transformation, 332

measure, 59 - absolutely continuous, 311 - complex, 365 - counting, 225 - discrete, 68 - Haar, 108 - induced by a transformation, 332 - inner regular, 103 - Lebesgue, 95 - Lebesgue-Stieltjes, 89, 115 - lower and upper variation of, 351 - outer, 73 - outer regular, 102 - signed, 345 - singular, 319 - total variation of a, 351 - total variation of, 368

measure space, 92 - complete, 92 - completeness of, 92 - completion of, 92

metric, L\, 41 Minkowski's inequality, 264 monotone,

- class, 86 - class generated, 86 - convergence theorem, 127

monotonicity property, 1

v < /i, 311, 356, 366 negative part of a

- function, 16, 49, 135 - signed measure, 351

negative set, 347 Neumann, John von, 315 Newton, Issac, 30 nonnegative simple measurable function, 118 norm, 159, 266

- induced by the inner product, 297 - of a bounded linear functional, 376 - of a complex measure, 371 - of a partition, 6

Index 417

- of a signed measure, 363 normed linear space, 266 null

- set, 22 - subset, 92

ft, 111 w ( / , J ) , 20 " ( / , * ) , 20 open intervals, 82, 96 orthogonal, 297

- complement, 298 oscillation of a function,

- at a point, 20 - in an interval, 20

outer measure, 73 - induced, 71 - Lebesgue, 95

outer regular, 102 outer regularity of L A, 102

p t h norm of / , 262 parallelogram identity, 297 Parseval's identity, 309 partial sum of the Fourier series, 307 partition,

- 6 - measurable, 129 - norm of, 6 - refinement of, 7 - regular, 14

Perron integral, 172 pointwise, 248 Poisson distribution, 69 polar,

- coordinate, 341 - coordinate transformation, 340 - representation, 372

popcorn function, 26 positive part,

- of a function, 16, 49, 135 - of a signed measure, 351

positive set, 347 power set, 55 probability, 94

- distribution function, 115 - measure, discrete, 68 - space, 94

product, - measure, 238 - measure space, 213, 238 - of measures (i and v, 213 - cr-algebra, 210, 212

projection theorem, 301 pseudo-metric, 43 Pythagoras identity, 298

(M,£ F , / x F ) , 314 R, 1 R e ( / ) , 243 .R-integrable, 17 n[a,b], 18 Radon, Johann, 44 Radon-Nikodym derivative, 320 Radon-Nikodym theorem,

- for complex measures, 367 - for finite measures, 354 - for measures, 319 - for signed measures, 357

random variable, 261 real part, 244 refinement of a partition, 7 regular

- measure, 324 - partition, 14, 15

regularity of AR2, 229 regularization of a function, 287 representation, standard, 118 Riemann, Bernhard, 32 Riemann

- integrable, 10 - integral, 10 - sum, 17

Riemann-Lebesgue lemma, 163 Riemann's example, 32 Riesz representation, 303 Riesz representation theorem, 377 Riesz theorem, 257 Riesz, Friedrich, 44 Riesz-Fischer theorem, 265, 308

S(PJ), 17 S ^ , 298 s± V s 2 , 120 si A s 2 , 120 5(C), 81 S*, 76 Saks' theorem, 97 section of E at x or y, 215 semi-algebra, 54 set function, 59

- countably additive, 59 - countably subadditive, 59 - finite, 84 - finitely additive, 59 - induced, 62 - monotone, 59 - cr-finite, 84

sigma algebra, 80 cr-algebra,

- generated, 81 - monotone class technique, 88 - monotone class theorem, 87 - of Borel subsets of R, 95

418

- of Borel subsets of X, 82 - product, 210 - technique, 82

cr-fmite, - set function, 84 - signed measure, 346

cr-set, 57 signed measure, 345 simple

- function, 48, 279 - function technique, 152 - measurable function, 135 - nonnegative measurable function, 118

singular - measure, 319 - value decomposition, 395 - values, 396

smaller, 365 smoothing of a function, 281 space,

- Banach, 266 - Hilbert, 297 - inner product, 297 - Lebesgue measure, 95 - measurable, 92 - measure, 92 - normed, 266 - probability, 94 - product measure, 213

spherical coordinates transformation, 344 standard representation, 118 Steinhaus theorem, 104 step function, 47, 50, 251 subspace, 298

- closed, 298 sum,

- lower or upper, 7 - Riemann, 17

supp ( / ) , 162 support, 162 symmetric moving average, 281

Theorem, - Arzela's, 40, 157 - bounded convergence, 151 - Darboux's, 18 - Egoroff's, 249 - Pubini, 189, 221, 222, 239 - Fundamental theorem of calculus, 34 - Hahn decomposition, 349 - Heine-Borel, 108 - inverse function, 405 - Jordan, 398 - Jordan decomposition, 350 - Lebesgue - Young, 179 - Lebesgue's dominated convergence, 148 - Luzin's, 254

Index

- monotone convergence, 127 - Riesz representation, 377 - Riesz-Fischer, 159, 308 - Saks', 97 - cr-algebra monotone class, 87 - Steinhaus, 104 - Ulam's, 61 - von Neumann, 315 - Vitali covering, 108

topological, - group, 108 - vector space, 261

total variation, 397 - of a complex measure, 368 - of a signed measure, 351

totally finite, 84 transition,

- measure, 229 - probability, 229

translation invariance, 4, 230 triangle inequality, 266 truncation sequence, 141

Ulam's theorem, 61 ultrafilter, 70 uncountable, 391 uniform distribution, 69 uniformly,

- absolutely continuous, 271 - integrable, 275

upper, - derivative, 327 - integral, 10 - left derivative, 180 - left limit, 180 - right derivative, 180 - right limit, 180 - sum, 7 - variation, 351

Urysohn's lemma, 280, 393

V?( / ) , 397 V j ( P , / ) , 397

variation, 397 - on R, 198

Vitali - cover, 108 - covering theorem, 108

Vitali's example, 113 Volterra's example, 37 Volterra, Vito, 37 von Neumann theorem, 315

Weirstrass, Karl, 36

(X,5,/Z), 92

Index of notations

Prologue

1 [0, +00]

Chapter 1

ll ll L(PJ) U(PJ) P1UP2

J a

I f(x)d: J a

I f(x)d J a

X

Set of real numbers, 1 Extended real numbers, 1 The collection of all intervals, The set {x G M * | x > 0 } , 1 Empty set, 1

Norm of a partition P , 6 Lower sum of / with respect to P , 6 Upper sum of / with respect to P , 7 Common refinement of P\ and P2, 7

Lower integral, 10

Riemann integral of / over [a, 6], 10

Upper integral, 10

Regular partition of an interval 14 Positive part of a function, 16 Negative part of a function, 16

419

420 Index of notations

S(PJ) K[a, b]

Chapter 2

Ln

Hi) i V(X) cnE r\E) HC)

M * <S *

S{C) Bx

BR

id M(C) CF HF

(X,S )

(X,S,/Z )

Riemann sum of / with respect to P , 17 Set of Riemann integrable functions on [a, 6], Oscillation of / in the interval J, 20 Oscillation of / at x, 20 Characteristic function of the set A, 30

18

: Collection of simple functions on R, 48

Chapter 3

The algebra generated by intervals, 52 The collection of left-open, right-closed intervals, 55 Power set of X; the collection of all subsets of X, 55 Subsets of E which are elements of C, 56 The set {x\f(x) G £ } , 57 The algebra generated by C, 57 Set function induced by the function F, 62 Outer measure induced by /i, 71 Collection of ^-measurable sets, 76 Sigma-algebra (cr-algebra) generated by C, 81 The a-algebra of Borel subsets of a topological space X, The cr-algebra of Borel subsets of R 82 Open intervals with rational end points, 82 Subintervals of [0,1] with dyadic end points, 82 Monotone class generated by C, 86 The cr-algebra of //^-measurable sets, 88 Lebesgue-Stieltjes measure induced by F, 89 Measurable space, 92 Measure space, 92 The completion of a measure space (X, 5, //),

82

92

Chapter 4

A* C A X0

diameter(F) fix) A + x xE

Lebesgue outer measure, 95 cr-algebra of Lebesgue measurable subsets of R, 95 The Lebesgue measure on R, 95 The collection of all open intervals in R, 96 The diameter of a subset E of R, 96 The derivative of / at x, 96 The set {y + x\y G A}, 101 The set {xy\y G E}, 103 Cardinality of the continuum, 110

Index of notations 421

ft

Chapter 5

: The first uncountable ordinal, 111 : Aleph nought, the cardinality of the set N, 112

U

/ sdji

S\ V 5 2

51 A 5 2

L+

fdji

The collection of nonnegative simple measurable functions, 118

Integral of a function sGLj" with respect to /i, 119

Maximum of the functions s\ and 52, 120 Minimum of the functions s\ and 52, 120 The class of nonnegative measurable functions, 123

Integral o f / G L + , 123 /

P a.e. x(n) on 7 ) n i n , , x , . _ , \ ^r > : r holds tor almost every x G Y P a.e. (ii)x G Y J J

with respect to /x, 125

/ fdfi : Integral of / over E with respect to /i, 125 JE

L L0

: The class of measurable functions, 135 : The class of simple measurable functions, 135

i(X,5,/x) 1 t

i(X), Li(/x) J ' The space of yu-integrable functions, 144

Li(E) Li[a,b]

C[a, b] supp(/)

a c°

C°°[a,b]

Cf POO

/ f{x)dx J a

The space of integrable functions on £", 154 The space of integrable functions on [a, 6], 154 The Li-norm of / , 158 The space of continuous functions on [a, 6], 160 Support of a function, 162 The space of continuous functions on R with compact support, 162 The space of infinitely differentiable functions on K, 164 The space of infinitely differentiable functions on [a, 6], 167 The space of functions in C°°(IR) with compact support, 168

Improper Riemann integral of / over [a, oo), 168

422 Index of notations

Chapter 6

liminf $(x) hie

limsup$(x) hie

liminf $(x)

limsup$(x) h^c

(D+f)(c) (D+f)(c) (D-f)(c) (D~f)(c)

Variation of / over the interval [a, 6], 176 Lower right limit of $ at c, 180

Upper right limit of <& at c, 180

Lower left limit of $ at c, 180

Upper left limit of $ at c, 180

Lower right derivative of / at c, 180 Upper right derivative of / at c, 180 Lower left derivative of / at c, 180 Upper left derivative of / at c, 180 Variation of F on R, 198

Chapter 7

A ® B : Product of the cr-algebra A with B, 211 BR2 : The cr-algebra of Borel subsets of R2, 211 (X x Y, A ® B, /J, x v) : The product measure space, 213

Ex

f2

detT

Section of E at x, 215 Section of E at y, 215 Lebesgue measurable subsets of R2, 229 Lebesgue measure on R2, 229 The set {/x J\I,J el}, 229 Determinant of T, 232

TT -X"i> ($0 A , TT /^ J : Product of a finite number

of measure spaces, 237 \i=\ 2=1

Afl£n

B^n S(x,r)

i= i

Lebesgue measurable subsets of Rn, 239 Lebesgue measure on Rn, 239 The cr-algebra of Borel subsets of Rn, 239 The open ball in Rn with center at x and radius r, 240

Index of notations 423

Chapter 8

C

R e ( / ) I m ( / ) L\(X,S,fi) Li(X,S,n)

Jn > J

Jn > J

fn^f /

a.u. p n y J

P m j , Jn • /

M(X)

Lp(X,S,n) Lp{n)

I I / I I P fh

f*9 Llfc(Rn) C°°{U) C?{U) 5 ( 0 , 1 ) Loo(X,5, / i ) ll/lloo C0(Rn)

L2(X,S,fx)

ll/lh (f,9) 5 X

C h a p t e r 9

(R,B F , / z F ) /i J_ z/

(£>/*)(*) (Dfi)(x) (D»)(x)

Field of complex numbers, 243 Real part of a complex-valued function / , 243 Imaginary part of a complex-valued function / , 244 Real-valued /^-integrable functions, 244 Complex-valued /x-integrable functions, 244

fn converges to / pointwise, 248

fn converges to / almost everywhere, 248

fn converges to / uniformly, 248

fn converges to / almost uniformly, 250

fn converges to / in measure, 255 The set of all measurable functions on X, 260

: The space of p t h-power integrable functions of / , 262

pth norm of / , 262 The function fh(x) := f(x + / i ) V x , 281 Convolution of / with g, 283 Space of locally integrable functions on Mn, 286 The set of infinitely differentiable functions on £/, 287 The set {/ G C°°(U) | supp( / ) is compact}, 287 Closure of the ball 5 ( 0 , 1 ) , 287 Space of essentially bounded functions, 291 Essential supremum of / , 292 The set of continuous functions on R n

vanishing at infinity, 295 Space of square integrable functions, 296 I/2-norm of / , 296 Inner product of / , g G L2, 296 The set {/ G L 2 ( X , S , M ) | < / , g) = 0 M g G 5 } , 298

v is absolutely continuous with respect to //, 311 Completion of the measure space (IR, Z3R, / ip) , 314 fi is singular with respect to v, 319

Radon-Nikodym derivative of v with respect to //, 320

Derivative of /i at x, 324 Upper derivative of \x at x, 327 Lower derivative of /i at x, 327

424 Index of notations

JT(x) :

Chapter 10

A * IA*I

v < //

-M6(X,<S ) IHI 2/ < /X

fi\/ u

/UL AlS

V < C /X

|z/| A4(X,<S)

Ml imi Appendices

The measure induced by T, Jacobian of T at x, 336

332

card(A)

No c 2 card(x)

A*

V*(PJ)

va\f)

(dT)(a)

JT{X)

Upper variation of a signed measure /x, 351 Lower variation of a signed measure /x, 351 Total variation of a signed measure /x, 351 Absolute continuity of a signed measure v with respect to a measure /x, 356

Integral of / with respect to a signed measure /x, 358

v absolutely continuous with respect to a signed measure /x, 359 Space of all finite signed measures on (X, <S), 362 Norm of a signed measure /x, 363 v is smaller than /x, 365 Maximum of /x and z , 365 Minimum of /x and z/, 365 Absolute continuity of a complex measure v with respect to a measure /x, 366 Total variation of a complex measure z/, 368 Space of all complex measures on (X, 5) , 371 Norm of a complex measure /x, 371 Norm of a bounded linear functional, 376

X equipotent to F , 391 Cardinality, cardinal number of A, 391 Cardinality of N, 391 Cardinality of the continuum, 391 Cardinality of the power set of X, 391 Transpose of a matrix, 395 Variation of / over [a, b] with respect to a partition P, 397 Variation of / over [a, 6], 397 Differential of a differentiate mapping T, j t h partial derivative of Ti at x, 401 Jacobian of T at x, 405

401

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